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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 10, OCTOBER 2006
A New Growing Method for Simplex-Based
Endmember Extraction Algorithm
Chein-I Chang, Senior Member, IEEE, Chao-Cheng Wu, Student Member, IEEE,
Wei-min Liu, Student Member, IEEE, and Yen-Chieh Ouyang, Member, IEEE
Abstract—A new growing method for simplex-based endmember extraction algorithms (EEAs), called simplex growing algorithm (SGA), is presented in this paper. It is a sequential algorithm
to find a simplex with the maximum volume every time a new
vertex is added. In order to terminate this algorithm a recently
developed concept, virtual dimensionality (VD), is implemented as
a stopping rule to determine the number of vertices required for
the algorithm to generate. The SGA improves one commonly used
EEA, the N-finder algorithm (N-FINDR) developed by Winter, by
including a process of growing simplexes one vertex at a time until
it reaches a desired number of vertices estimated by the VD, which
results in a tremendous reduction of computational complexity.
Additionally, it also judiciously selects an appropriate initial vector to avoid a dilemma caused by the use of random vectors as its
initial condition in the N-FINDR where the N-FINDR generally
produces different sets of final endmembers if different sets of
randomly generated initial endmembers are used. In order to
demonstrate the performance of the proposed SGA, the N-FINDR
and two other EEAs, pixel purity index, and vertex component
analysis are used for comparison.
Index Terms—Endmember extraction, N-finder algorithm
(N-FINDR), pixel purity index (PPI), sequential endmember extraction algorithm (SQEEA), simplex growing algorithm (SGA),
simultaneous endmember extraction algorithm (SMEEA), vertex
component analysis (VCA), virtual dimensionality (VD).
I. I NTRODUCTION
E
NDMEMBER extraction has become increasingly important in hyperspectral image analysis due to significantly
improved high spatial and spectral resolution provided by hyperspectral imaging sensors. According to the definition given
in [1], an endmember is an idealized, pure signature for a class.
For multispectral imagery, an endmember may be difficult to
find since most image pixels are heavily mixed because of low
spatial and spectral resolution. As a result, the importance of
endmember extraction has been overlooked, and its issue has
Manuscript received October 1, 2005; revised July 12, 2006.
C.-I Chang is with the Remote Sensing Signal and Image Processing Laboratory, Department of Computer Science and Electrical Engineering, University
of Maryland, Baltimore County, Baltimore, MD 21250 USA, and also with
the Department of Electrical Engineering, National Chung Hsing University,
Taichung 402, Taiwan, R.O.C. (e-mail: [email protected]).
C.-C. Wu and W. Liu are with the Remote Sensing Signal and Image Processing Laboratory, Department of Computer Science and Electrical Engineering, University of Maryland, Baltimore County, Baltimore, MD 21250 USA
(e-mail: [email protected]; [email protected]).
Y.-C. Ouyang is with Department of Electrical Engineering, National Chung Hsing University, Taichung 402, Taiwan, R.O.C. (e-mail:
[email protected]).
Digital Object Identifier 10.1109/TGRS.2006.881803
not been a major subject in multispectral image analysis. By
contrast, with advances of hyperspectral imaging sensors many
subtle material substances that cannot be resolved by multispectral imagery can now be uncovered by hyperspectral imagery.
These substances are generally not known a priori and can be
only diagnosed by high spectral resolution. Endmembers are
considered to one type of such substances where their existence
in image data cannot be detected visually. Most importantly,
once endmembers are present, they may generally appear as
anomalies since their population is relatively small. Because
of such characteristics, finding endmembers is very challenging. Many algorithms have been developed for this purpose,
such as the pixel purity index (PPI) [2], N-finder algorithm
(N-FINDR) [3], iterative error analysis (IEA) [4], automated
morphological endmember extraction (AMEE) algorithm [5],
minimum volume transform (MVT) [6], convex geometry [7],
convex cone analysis (CCA) [8], vertex component analysis
(VCA) [9], which can be categorized into two classes, simultaneous endmember extraction algorithms (SMEEAs) including
PPI, N-FINDR, MVT, CCA, convex geometry, and sequential
endmember extraction algorithms (SQEEAs) including IEA,
AMEE, and VCA. Technically speaking, an optimal endmember extraction algorithm (EEA) must be an SMEEA. This is
because all the endmembers should be selected all together at
one time rather than one after another sequentially. However,
finding simultaneously endmembers generally requires tremendous computational complexity due to exhaustive search. On
the other hand, despite the fact that an SQEEA may not be
as optimal as an SMEEA can be, a well-designed SQEEA
may be able to perform as well as an SMEEA can. The most
advantage benefited from an SQEEA is the significant reduction
of computation complexity.
A key idea to design an EEA is to use convexity of the
data structure. One such approach is simplex-based methods,
which find an appropriate set of vertices that represents desired
endmembers in some optimal sense. Among these algorithms
is the widely used N-FINDR developed by Winter [3], which
finds a simplex of the maximum volume with a given number
of vertices p. It is based on an assumption that for a given
(p − 1)-dimensional simplex, the simplex that yields the largest
volume will be the one whose p vertices are most likely
specified by purest pixels. The vertices of an N-FINDR-found
simplex are the desired set of endmembers. Unfortunately,
there are several disadvantages of implementing the N-FINDR.
One is that there is no provided criterion to determine how
many endmembers for the N-FINDR to generate. Another is
0196-2892/$20.00 © 2006 IEEE
CHANG et al.: NEW GROWING METHOD FOR SIMPLEX-BASED ENDMEMBER EXTRACTION
that the N-FINDR uses randomly generated vectors as initial
endmembers, which are not an effective way to initialize the
algorithm. It generally takes a long time to find a desired set
of endmembers. Most importantly, due to the nature in the
use of random initial endmembers the N-FINDR generally
produces different sets of final endmembers at separate runs.
Accordingly, how many runs that the N-FINDR must be
performed and which set of final generated endmembers by the
N-FINDR should be selected as a desired set of endmembers
become issues in endmember extraction despite that many
of them may be overlapped. Additionally, it is an SMEEA
and requires enormous computation to conduct an exhaustive
search to find its final set of endmembers. Inspired by the
N-FINDR and its drawbacks this paper presents a simplex
growing algorithm (SGA) that can be used to resolve these
issues commonly encountered in simplex-based algorithms.
As for the first issue in determination of the number of
endmembers required to be generated p, there is no guideline
suggested in many EEAs [2]–[8]. This issue has been left
open. In the SGA, a newly developed concept, called virtual
dimensionality (VD) [10], [11], which has recently shown
success in determination of the number of endmembers for
endmember extraction [9], [12], [13], is proposed to estimate
such p. Most recently, a linear mixture model-based least
squares error method was proposed to estimate signal subspace
in hyperspectral imagery [14]. This method, referred to as
signal subspace estimation (SSE) in this paper, can be also used
to estimate p. Once p is determined, the N-FINDR starts with
a set of p random initial vertices and repeatedly calculates the
volumes of new simplexes with new vertex replacements until
it finds a simplex with the largest volume. Since the N-FINDR
is an SMEEA, the entire process must be repeated over again
when each replacement is taken place. Therefore, if the number
p is large, the N-FINDR becomes very slow. Our proposed
SGA takes a rather different approach. It finds a desired
(p − 1)-dimensional simplex with the largest volume by gradually growing simplexes vertex by vertex. In other words, instead
of making an attempt to directly find a p-vertex simplex with
the maximum volume, it first finds a two-vertex simplex with
the largest volume from which it begins to grow new simplexes
with the largest volumes by increasing vertices from 2 to p.
Since it generates desired endmembers one by one through a
simplex growing process, the SGA is an SQEEA. With such
a simplex growing implementation, the SGA only has to find
one endmember at a time until it reaches a desired number of
endmembers, which is the VD-estimated p. This is completely
different from the N-FINDR, which is an SMEEA and replaces
vertices of simplexes with a number of newfound vertices
repeatedly. A third issue for the N-FINDR is a consequence
resulting from its use of random vectors as initial endmembers.
If the N-FINDR is rerun again, it is most likely that a different
set of final endmembers is generated due to the use of a
different set of random initial vectors. The SGA resolves this
issue by judiciously selecting the initial vectors to initialize the
algorithm. Accordingly, the final generated set of endmembers
is always the same and consistent.
Incidentally, a recent VCA approach developed in [9] is
similar to the SGA from two aspects. Both use the VD to
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estimate the number of endmembers required to be generated.
Besides, both also grow simplexes gradually vertex by vertex to
find desired endmembers. However, there are also two distinct
features between the VCA and the SGA. The VCA repeatedly
performs orthogonal subspace projections resulting from a
sequence of gradual growing simplexes vertex by vertex to find
new vertices. By contrast, the SGA finds maximum volumes
for a sequence of gradual growing simplexes vertex by vertex.
Furthermore, the SGA develops a specific algorithm to generate
initial endmembers so that the final selection of endmembers by
the SGA is consistent regardless of how many runs the SGA is
implemented. But, like the N-FINDR, the VCA makes use of a
zero-mean Gaussian distribution to generate a random vector as
its initial projection for each simplex it grows. As a result, the
VCA suffers from the same drawback as does the N-FINDR.
That is, final sets of endmembers generated by the VCA in
different runs are usually not consistent.
In order to demonstrate the performance of the proposed
SGA, a comparative analysis among the SGA, PPI, N-FINDR,
and VCA is conducted via synthetic image-based computer
simulations as well as two real hyperspectral image data where
experimental results show that the SGA generally performs
significantly better than the N-FINDR in terms of resolving the
issues addressed above and also comparable or slightly better
than the VCA.
The remainder of this paper is organized as follows.
Section II describes an implementation of the N-FINDR developed by Winter [3]. Section III presents a new simplex
growing method for endmember extraction, called an SGA.
Section IV compares the computational complexity among the
PPI, VCA, N-FINDR, and SGA. Sections V and VI conduct
a comparative study among the PPI, N-FINDR, VCA, and
SGA via synthetic image-based computer simulations and real
hyperspectral image experiments for performance analysis and
evaluation. Section VII makes concluding remarks.
II. N- FINDER A LGORITHM
The main idea of N-FINDR is to assume that a (p − 1)dimensional volume formed by a simplex with p vertices that
are specified by the purest pixels is always larger than that
formed by another combination of p pixels. N-FINDR was
briefly described in [3]. Unfortunately, a detailed step-by-step
algorithmic implementation was not provided in [3]. In this section, we summarize the steps to implement N-FINDR according
to our understanding and experience as follows. It by no means
claims that our interpreted N-FINDR is identical to the one
developed by Winter [3]. Nevertheless, the idea used in both
algorithms should be the same.
N-FINDR
1) Preprocessing:
a) let p be the number of endmembers required to generate; and
b) apply a maximum noise fraction (MNF) transformation [or called noise-adjusted principal component
(NAPC)] to reduce the data dimensionality from L to
p − 1 where L is the total number of spectral bands.
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(0)
(0)
(0)
2) Initialization: Let {e1 , e2 , . . . , ep } be a set of initial
vectors randomly generated from the data.
(k) (k)
(k)
3) At iteration k ≥ 0, find V (e1 , e2 , . . . , ep ) defined by
(k)
V e1 , . . . , e(k)
=
p
1
det (k)
e
1
...
...
(p − 1)!
1
(k)
e2
1
(k)
ep
(1)
which is the volume of the simplex with vertices
(k) (k)
(k)
(k) (k)
(k)
e1 , e2 , . . . , ep , denoted by S(e1 , e2 , . . . , ep ).
4) Stopping rule: For each sample vector r, we re(k)
(k)
(k)
(k)
(k)
calculate V (r, e2 , . . . , ep ), V (e1 , r, e3 , . . . , ep ),
(k)
(k)
. . . , V (e1 , . . . , ep−1 , r), the volumes of p simplex(k)
(k)
(k)
(k)
(k)
es, S(r, e2 , . . . , ep ), S(e1 , r, e3 , . . . , ep ), . . . ,
(k) (k)
(k)
S(e1 , e2 , . . . , ep−1 , r), each of which is formed
(k)
with the samby replacing one endmember ej
ple vector r. If none of these p recalculated vol(k)
(k)
(k)
(k)
(k)
umes, V (r, e2 , . . . , ep ), V (e1 , r, e3 , . . . , ep ),
(k)
(k)
(k)
. . . , V (e1 , . . . , ep−1 , r), is greater than V (e1 ,
(k)
(k)
(k)
(k)
(k)
e2 , . . . , ep ), no endmember in e1 , e2 , . . . , ep
will be replaced. The algorithm is terminated. Otherwise,
continue.
5) Replacement rule: The endmember pixel which is
absent in the largest volume among the p sim(k)
(k)
(k)
(k)
(k)
plexes, S(r, e2 , . . . , ep ), S(e1 , r, e3 , . . . , ep ),
(k) (k)
(k)
. . . , S(e1 , e2 , . . . , ep−1 , r), will be replaced by the
sample vector r. Assume that such an endmember is
(k+1)
. A new set of endmembers is then
denoted by ej
(k+1)
(k+1)
produced by letting ej
= r and ei
i = j. Let k ← k + 1 and go to step 3).
(k)
= ei
for
III. S IMPLEX G ROWING A LGORITHM
Since an endmember is an idealized pure signature, it is
not necessarily an image pixel. However, in real-image experiments, an endmember is generally extracted directly from the
data. Therefore, when it occurs as a pixel, it is referred to as
endmember pixel in this paper.
In this section, we present a new algorithm, called SGA, for
endmember extraction to find a set of desired endmembers by
growing a sequence of simplexes. It starts off with two vertices
and begins to grow a simplex by increasing its vertices one at a
time. The algorithm is terminated when the number of vertices
reaches the number of endmembers p, which can be estimated
by the VD using a method developed by Harsanyi, Farrand,
and Chang, referred to as HFC method in [10], which does not
require noise estimation. In order to select an appropriate pixel
as its initial endmember pixel, a selection process for finding
the first endmember pixel is developed for this purpose and
described as follows.
First Endmember Selection Process
1) Randomly generate a target pixel, denoted by t.
of absolute
2) Find a pixel e1 that yields
the maximum
1 1 over all sample
determinant of the matrix, det
t r vectors r, i.e.,
1 1 e1 = arg max det
t r r
where principal components analysis (PCA) or MNF is
required to reduce the original data dimensionality L to
the dimension 2 to find the maximum.
It is worth noting that the generation of the first endmember
pixel e1 is determined by the randomly generated target pixel t.
A different target pixel t may result in a different e1 . Interestingly, the experiments show that the generated e1 is always
a pixel which has either a maximum or a minimum value in
the first component of dimensionality reduction (DR) transform. Therefore, the target pixel t has no effect on the final
set of endmembers. Furthermore, according to our extensive
experiments, the generated e1 eventually becomes one of the
final generated endmembers. This explains that the final set
of endmembers generated by the SGA is always the same and
consistent.
Including the above first endmember selection process as a
preprocessing for initialization, the SGA can be described in
detail as follows.
Simplex Growing Algorithm
1) Initialization:
a) use the VD to estimate the number of endmembers
to be generated p; and
b) use the e1 found by the first endmember selection
process as the desired initial endmember pixel and
set n = 1.
2) At n ≥ 1 and for each sample vector r, we calculate
V (e1 , . . . , en , r) defined by
det 1 1 . . . 1 1 e1 e2 . . . en r V(e1 , . . . , en , r) =
(2)
n!
which is the volume of the simplex specified by vertices
e1 , e2 , . . . ,en , r, denoted by S(e1 , e2 , . . . , en , r). Since
1 1 ... 1 1
the matrix
in (2) is not necessare1 e2 . . . en r
ily a square matrix, a DR technique such as PCA or MNF
is required to reduce the original data dimensionality L to
the dimension n.
3) Find en+1 that yields the maximum of (2), i.e.,
en+1 = arg max [V (e1 , . . . , en , r)] .
(3)
r
4) Stopping rule: If n < p, then n ← n + 1 and go step 2).
Otherwise, the final set of {e1 , e2 , . . . , ep } is the desired
p endmembers.
CHANG et al.: NEW GROWING METHOD FOR SIMPLEX-BASED ENDMEMBER EXTRACTION
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TABLE I
COMPUTATIONAL COMPLEXITY AMONG PPI, N-FINDR, VCA, AND SGA
Fig. 2. Five USGS ground-truth mineral spectra.
Fig. 1.
Plots of numbers of flops versus the number of endmembers, p.
IV. C OMPUTATIONAL C OMPLEXITY
In this section, a comparative study is conducted on the
computational complexity among PPI, VCA, N-FINDR, and
SGA. Since the N-FINDR uses randomly generated initial
endmembers, it generally results in different numbers of replacements for different runs. Therefore, in this paper, the
best N-FINDR was used to compare against the other three
algorithms where a best N-FINDR represents a case that no
replacement is required for the initial endmembers, in which
case the initial endmembers turned out to be the final endmembers. According to the calculation of computational complexity
used in [9], approximate numbers of flops (i.e., numbers of
floating operations) required for the four EEAs are tabulated in
Table I where N is the total number of pixels, p is the number
of endmembers, and s is the number of skewers.
Since the computational complexity of performing DR and
VD was relatively small and negligible, it was not included
in Table I. A best N-FINDR computed the determinant of a
p × p matrix N p times and the computational complexity of
each time was pη with 2.3 < η < 2.9. The SGA computes the
determinant of a matrix N n times with n starting from 2 to
p. The VCA projected a total of N p-dimensional data vectors
onto p orthogonal projections. The PPI projects all data vectors
onto a large number of skewers s. Fig. 1 plots the number
of flops versus p for each of the four considered EEAs with
N = 104 and s = 103 . As shown in Fig. 1, the number of flops
required for the SGA was always less than that required for a
best N-FINDR where the VCA had the lowest numbers.
As a concluding remark, a note is worthwhile. In terms
of operations implemented in an EEA, the SGA and the
N-FINDR as a group, which perform computations of simplex
volumes as opposed to the PPI and the VCA as another group,
which performs orthogonal projections. Since these two oper-
Fig. 3. (a) Twenty-five simulated panels. (b) A synthetic image having the
25 panels simulated in (a) implanted in the background with an additive
Gaussian noise to achieve SNR 20 : 1.
ations are different in nature, it may not be a fair comparison
between these two groups. However, Fig. 1 shows that within
each group, the SGA and VCA were significantly better than
the N-FINDR and PPI, respectively. Nevertheless, it should
be noted that computing an orthogonal projection is generally
faster than that calculating a simplex volume. Accordingly, the
VCA yielded the lowest computational cost if the performance
is only measured by computing time. As noted, since our
proposed SGA is improved from the N-FINDR, it is important
to conduct a comparative study on computational complexity
between these two. The computation complexity of the PPI and
VCA is included in our paper only for reference.
V. C OMPUTER S IMULATIONS
In order to conduct a comprehensive comparative analysis, a
synthetic image was custom-designed and simulated based on
the reflectance spectra of five U.S. Geological Survey (USGS)
ground-truth mineral spectra: Alunite (A), Buddingtonite (B),
Calcite (C), Kaolinite (K), and Muscovite (M) shown in Fig. 2.
It should be noted that the simulated synthetic image may not
be realistic from practical applications. But it does allow us to
simulate various scenarios to explore many interesting insights
that cannot be observed from real image experiments.
The synthetic image has size of 200 × 200 pixel vectors
with 25 panels of various sizes that are arranged in a 5 × 5
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TABLE II
SIMULATED 20 MIXED PANEL PIXELS IN THE THIRD COLUMN
TABLE III
ABUNDANCE FRACTIONS OF SUBPIXEL PANELS IN THE
FOURTH AND FIFTH COLUMNS
Fig. 4. Results of PPI with two different runs using MNF and 50 skewers.
(a) VD = 6. (b) SSE = 5.
matrix and located at the center of the scene shown in Fig. 3(a).
The five mineral spectral signatures, {mi }5i=1 in Fig. 2 were
used to simulate these 25 panels where each row of five panels
was simulated by the same mineral signature and each column
of five panels has the same size. Among 25 panels are five
4 × 4 pure pixel panels for endmember extraction, pi4×4 for
i = 1, . . . , 5 in the first column, five 2 × 2 pure pixel panels for
training samples, pi2×2 for i = 1, . . . , 5 in the second column,
five 2 × 2 mixed pixel panels, {pi3,jk }2,2
j=1,k=1 for i = 1, . . . , 5
in the third column for mixed pixel classification, five subpixel
panels, pi4,1 for i = 1, . . . , 5 in the fourth column for subpixel
classification and five subpixel panels, pi5,1 for i = 1, . . . , 5 in
the fifth column for subpixel classification. The reason that
the five panels in the second column were included in the
image scene was to use them as training samples for supervised
classification. The purpose of introducing the five panels in the
third column was designed to conduct a study and analysis
on five mineral signatures with different mixing in a pixel.
Table II tabulates the mixing details of mineral composition in
the 20 panels.
The inclusion of the panels in the fourth and fifth columns
is to investigate subpixel effect on endmember extraction and
their simulated abundance fractions are tabulated in Table III
where the background (BKG) was simulated by mixing 20%
of each of five mineral signatures, A, B, C, K, and M, i.e.,
20%A + 20%B + 20%C + 20%K + 20%M.
These 25 panels in Fig. 3(a) were implanted in the image
background in a way that the background pixels were replaced
with the implanted panel pixels. Finally, this synthetic image
with the 25 implanted panels was corrupted by a simulated
white Gaussian noise to achieve signal-to-noise ratio 20 : 1
defined in [10] and [15] as 50% reflectance divided by the
standard deviation of the noise. The resulting noisy synthetic is
shown in Fig. 3(b). For the synthetic image scene in Fig. 3(b),
there are 100 pure pixels, 20 mixed pixels, and 10 subpixels,
all of which were simulated by five distinct pure mineral
signatures. The VD estimated for this synthetic image was 6
with PF ≤ 10−1 that include five mineral signatures, A, B, C,
K, M, and the background signature made up of mixing 20% of
each minerals, A, B, C, K, and M.
It should be noted that although the background is equally
mixed by five minerals, it is indeed a signature spectrally
distinct from the five minerals. This may be due to the fact
that the simulated background signature is regarded as a hybrid
signature in its own instead of being considered as a mixed
signature. As a result, this hybrid signature becomes a new endmember that represents the background class. In this case, the
VD counted it as one endmember. Interestingly, a linear mixture
model-based least squares error technique recently developed
in [14], called SSE can be also used for this purpose where
p was estimated to be p = 5. In order to see its performance,
the results using SSE = 5 were also included in comparison
with VD = 6. With the number of endmembers, p estimated by
VD = 6 and SSE = 5, two types of EEAs, SMEEA (PPI and
N-FINDR), and SQEEA (VCA and SGA) were implemented
for comparative analysis.
This experiment is to show a dilemma resulting from the
use of randomly generated initial endmembers. That is, final
endmembers selected by an EEA are inconsistent. Figs. 4–6
show the final selected endmembers by PPI, N-FINDR, and
VCA in two different runs, respectively, where the value of p
was estimated by VD and SSE as VD = 6 and SSE = 5. The
PPI implemented here was the one described in [2] with number
of skewers chosen to be 50.
The N-FINDR found six pixels shown in Fig. 5 where one
pixel from each of five 4 × 4 panels in the first column
marked by open circles was extracted in correspondence to
CHANG et al.: NEW GROWING METHOD FOR SIMPLEX-BASED ENDMEMBER EXTRACTION
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Fig. 7. Results of SGA using MNF. (a) VD = 6. (b) SSE = 5.
Fig. 5. Results of N-FINDR with two different runs using MNF. (a) VD = 6.
(b) SSE = 5.
Fig. 6.
Results of VCA in two different runs. (a) VD = 6. (b) SSE = 5.
five pure signatures, A, B, C, K, M, and one background pixel
corresponding to a mixed signature was extracted. It should be
noted that since the PPI and N-FINDR are SMEEAs, its six
endmember pixels were generated simultaneously. Therefore,
no numbers were shown in Figs. 4 and 5.
Due to the use of insufficient number of skewers, PPI missed
one endmember that represents the signature of the mineral
Calcite, but it could be improved to extract all the five endmembers if 500 skewers were used [16]. Additionally, despite
that the results shown in Fig. 4 in two different runs looked the
same, their PPI counts were actually different.
As shown in Figs. 5–7, all three N-FINDR, VCA, and SGA
successfully extracted five pixels that were specified by the
five pure signatures, A, B, C, K, and M plus one different
background pixel that was not an endmember pixel. As also
shown in Fig. 6, the background pixels extracted by the VCA
in two different runs were different. Additionally, Fig. 6(a)
and (b) further shows that the order of the five endmembers
corresponding to the five pure signatures, A, B, C, K, M
extracted by the VCA in two different runs was also different.
This further demonstrated that the results produced by the VCA
in different runs were not consistent.
Additionally, Figs. 6 and 7 show the endmember pixels found
by the VCA and SGA in two different runs using p = 5 and 6,
respectively, where the numbers indicated the order of six
pixels extracted by the VCA and the SGA in sequence and
a background signature was always extracted prior to the last
extracted endmember. As a result, when p = 5 estimated by the
SSE was used, none of the four studied EEAs could extract all
five distinct mineral endmembers. They always missed the third
panel signature in the first column. On the other hand, if p = 6
estimated by the VD was used, all the four EEAs successfully
extracted all five minerals. These experiments demonstrated
that in order for an EEA to extract all necessary five minerals,
the minimum value chosen for p must be at least six estimated
by the VD, not five estimated by the SSE.
Since all the four EEAs require DR as a preprocessing prior
to endmember extraction. In order to see the impact of different
transforms used for DR, three commonly used transforms,
PCA, MNF, and independent component analysis (ICA), were
implemented to conduct comparative analysis. It should be
noted that a problem similar to the use of random initial endmembers in the PPI, N-FINDR, and VCA is also encountered in
the ICA implementation, which also uses randomly generated
projectors as its initial conditions. In order to mitigate this
dilemma, a recent work developed for DR in [17] was implemented as the ICA as DR transform in this paper. Figs. 8–10
show the results of the PPI, N-FINDR, and SGA with VD = 6
and SSE = 5 where the number of dimensions to be retained
after DR was set to be the same as p. It should be noted that no
experiments were conducted for the VCA since the algorithm
used to run the VCA has its built-in DR transform [9].
By examining the results in Figs. 6 and 8–10, the four EEAs
extracted all the five minerals for VD = 6 and missed one
mineral for SSE = 5 regardless of which DR transform was
used. This implied that the three DR transforms made very little
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Fig. 8. Endmember pixels extracted by PPI using three different DR transforms. (a) VD = 6. (b) SSE = 5.
Fig. 10. Endmember pixels extracted by SGA using three different DR
transforms. (a) VD = 6. (b) SSE = 5.
TABLE IV
COMPUTING TIME OF PPI, N-FINDR, VCA, AND SGA
TABLE V
COMPUTER ENVIRONMENTS USED FOR THE N-FINDR, VCA, AND SGA
Fig. 9. Endmember pixels extracted by N-FINDR using three different
DR transforms. (a) VD = 6. (b) SSE = 5.
“for” loops than VCA does. Additionally, calculating simplex
volume as performed in the SGA generally requires much more
computing time than projection carried out in the VCA. It also
showed that an SQEEA (i.e., VCA, SGA) is generally much
more efficient than an SMEEA (i.e., PPI, N-FINDR).
Three concluding remarks are noteworthy.
difference in endmember extraction. However, this conclusion
will no longer be true as demonstrated in real-image
experiments in Section VI. From the above experiments,
the only major difference was the value of p for an EEA to
generate, in which case VD seemed to provide a better estimate
than SSE did.
Table IV tabulates the computing time required for PPI,
N-FINDR, VCA, and SGA with VD = 6 using computer
environments described in Table V and MATLAB 7.02 service
pack 2 where VCA had the best computing time, which
was about four times faster than SGA, which was in turn
four times faster than N-FINDR where VCA was the most
efficient one. However, it is worth noting that in comparison
of VCA with SGA their computing time did not truly reflect
computational complexity. This is because MATLAB performs
“matrix” operations much faster than “for” loops (that is why
it is named after Matrix Lab) and the SGA has many more
1) A prominent difference between SMEEA and SQEEA
is that the former searches all the p endmembers simultaneously, while the latter looks for an endmember at
a time sequentially. As a result, an SMEEA generally
requires an exhaustive search that results in a very high
computational complexity. Therefore, to avoid such expensive computing cost, many SMEEAs are actually designed and developed by searching desired endmembers
in feasible ranges rather than the entire image. N-FINDR
is one which implements step 3) iteratively to replace
unlikely endmembers among those already found in a
previous iteration with more likely pixels in the current iteration. Therefore, technically speaking, N-FINDR is not
an optimal SMEEA. If it is, N-FINDR must replace all the
p pixels at a time and find one set of p pixels that yields
the maximum simplex volume. In this case, the final p
pixels would be desired endmember pixels. Additionally,
CHANG et al.: NEW GROWING METHOD FOR SIMPLEX-BASED ENDMEMBER EXTRACTION
the controversy in final inconsistent results resulting from
the use of randomly generated initial endmembers can be
further avoided because the search is conducted exhaustively in the entire image regardless of which p pixels
are picked for processing initially.
Unfortunately, such an
exhaustive search requires Np = N !/(N − p)!p! combinations of p pixels. When N becomes large, the searching process also becomes forbidden. This evidence is
witnessed by our implementation of CCA [8] for the
above synthetic image. Since it is an exhaustive search
algorithm, it took tens of hours using the same computer
environment implemented by other EEAs to produce the
results similar and close to the results presented in this
section except different background pixels. Despite that
the CCA is of major theoretical interest, it is impractical
for applications unless computer environments can be
significantly improved to the point that computations are
not a major hurdle. Therefore, it is unrealistic to include
the CCA experiments in our computational complexity
analysis. On the other hand, an SQEEA overcomes this
dilemma by producing one endmember at a time sequentially. As a result, its computational complexity can be
tremendously reduced. Of course, it is also not an optimal
algorithm. Nevertheless, according to our experiments,
an SQEEA nearly works as well and effectively as an
SMEEA does.
2) As noted in the above experiments, the value of p estimated by the VD and SSE was close, 6 for VD and 5 for
SSE. However, in our synthetic image-based simulations,
even though SSE differed from the VD by one, the
difference between final results in endmember extraction
was significant because all the four studied EEAs missed
one endmember if p = 5. For two SMEEAs, the PPI and
N-FINDR missed panels in the third row. For two
SQEEAs, the order of the extracted endmembers showed
that the VCA and SGA always extracted one background
pixel before they extracted the last endmember. In other
words, the last extracted endmember always happened
to be the sixth pixel. This implies that the background
signature must be considered as one of endmembers. This
fact was observed in Figs. 5–7 where the background
signature was always among the first five extracted signatures, while the fifth mineral signature was always the last
and the sixth signature extracted by the N-FINDR, VCA,
and SGA. This implies that in order to extract all the
five mineral signatures, the least number of endmembers
for all the three considered EEAs, N-FINDR, VCA, and
SGA to generate must be 6 not 5. In other words, if
the number of endmembers is 5, no matter which EEA
is implemented, the algorithm always extracts four mineral signatures and one background signature with one
missing mineral signature. Therefore, the VD actually
provided a more accurate estimate of p than the SSE did.
It is also worth noting that according to the definition of
endmember given in [1], an endmember is intended to be
used to represents a class. In light of this interpretation,
VD = 6 is an accurate estimate since there are six classes
in the synthetic image in Fig. 3(b), which include five
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Fig. 11. (a) HYDICE panel scene that contains 15 panels. (b) Ground truth
map of spatial locations of the 15 panels.
mineral classes plus a background class. This is because
the signature that specifies the background class can be
considered as a hybrid signature rather than a mixed
signature. Furthermore, it can be easily shown that if the
OSP classifier in [15] was used to classify the five mineral
signatures in Fig. 3(b), using six signatures to form the
signature matrix performed better than using only five
signatures since the background signature must be used
to account for an additional class for background removal
so as to achieve better classification.
3) It is our belief that there are two major issues for the SSE
to fail in the above experiments. One is that the SSE is
very sensitive to the estimated noise covariance matrix.
As a matter of fact, the noise estimation on SSE = 5
is based on multiple regression theory as in [18]. The
other issue is the linear mixture model-based least squares
error approach used by the SSE. If a hyperspectral image
cannot be well represented by a linear mixture model,
then it can be expected that the SSE may be very likely
to fail. The advantage of the VD over the SSE is that the
VD is only determined by the false alarm probability, not
by the linear mixture model, nor by noise estimates.
VI. R EAL -I MAGE E XPERIMENTS
Two real-image scenes collected by the Hyperspectral Digital
Image Collection and Experiment (HYDICE) and Airborne
Visible Infrared Imaging Spectrometer (AVIRIS) were used for
experiments. The four algorithms, PPI, N-FINDR, VCA, and
SGA, were evaluated for performance analysis.
A. HYDICE Data
The first image data to be studied consist of an image scene
shown in Fig. 11(a), which has a size 64 × 64 pixel vectors with
15 panels in the scene and the ground truth map in Fig. 11(b)
in [10]. It was acquired by 210 spectral bands with a spectral
coverage from 0.4 to 2.5 µm. Low signal/high noise bands
bands 1–3 and bands 202–210; and water vapor absorption
bands: bands 101–112 and bands 137–153 were removed.
Therefore, a total of 169 bands were used in experiments. The
spatial resolution and spectral resolution of this image scene are
1.56 m and 10 nm, respectively. Within the scene in Fig. 11(a)
there is a large grass field background, and a forest on the
left edge. Each element in this matrix is a square panel and
denoted by pij with rows indexed by i and columns indexed by
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TABLE VI
VD ESTIMATES FOR THE HYDICE SCENE IN Fig. 11 WITH
VARIOUS FALSE ALARM PROBABILITIES
j = 1, 2, 3. For each row i = 1, 2, . . . , 5, there are three panels
painted by the same paint but with three different sizes. The
sizes of the panels in the first, second, and third columns are
3 × 3 m, 2 × 2 m, and 1 × 1 m, respectively. Since the size
of the panels in the third column is 1 × 1 m, they cannot be
seen visually from Fig. 11(a) due to the fact that its size is less
than the 1.56-m pixel resolution. For each column j = 1, 2, 3,
the five panels have the same size but with five different paints.
However, it should be noted that the panels in rows 2 and 3 were
made by the same material with two different paints. Similarly,
it is also the case for panels in rows 4 and 5. Nevertheless, they
were still considered as different panels but our experiments
will demonstrate that detecting panels in row 5 (row 4) may also
have effect on detection of panels in row 2 (row 3). The 1.56-m
spatial resolution of the image scene suggests that most of the
15 panels are one pixel in size except that the panels in the first
column with the second, third, fourth, fifth rows, which are twopixel panels, denoted by p211 , p221 , p311 , p312 , p411 , p412 , p511 ,
p521 . Fig. 11(b) shows the precise spatial locations of these
15 panels where red pixels (R pixels) are the panel center pixels
and the pixels in yellow (Y pixels) are panel pixels mixed with
the background.
First of all, the VD was used to estimate number of bands
p required for band selection. Table VI calculated the values
of VD for the HYDICE image in Fig. 11(a) with various false
alarm probabilities.
For our experiments, VD was chosen to be 9. The selection of
p = 9 is empirical based on the false alarm fixed at probabilities
PF = 10−3 , 10−4 . Once again, the value of p estimated by the
SSE was 10.
Experiments similar to those conducted for the synthetic
image in Section V were also performed in this section. First
of all, we demonstrated inconsistent results of endmembers
extracted by the PPI, N-FINDR, and VCA using two different
sets of randomly generated initial endmember pixels in two separate runs. Figs. 12–14 show the results of endmember pixels
extracted and marked with circles by the three EEAs in two
different runs, respectively, where the value of p was estimated
by the VD = 9 and SSE = 10 and both PPI and N-FINDR
used the MNF for DR. It should be noted that the PPI used
1000 skewers to extract endmember pixels in Fig. 12. For
p = 9 estimated by the VD, the PPI produced the four panel
pixels p11 , p311 , p412 , and p521 in one run and another three
panel pixels p11 , p511 , and p521 with two panel pixels different
which are p312 and p412 . All the four panel pixels represented
four distinct panel signatures, p1 , p3 , p4 , p5 as endmembers
but their PPI counts were different. On the other hand, for
p = 10 estimated by the SSE, the PPI only extracted three panel
pixels, p11 , p412 , and p521 in one run and three panel pixels,
p11 , p312 , and p521 in another run. All the extracted panel
pixels represented three distinct panel signatures, p1 , p4 , p5
as endmembers which were one short of four endmembers with
Fig. 12. Endmember pixels extracted by PPI with MNF in two different runs.
(a) VD = 9. (b) SSE = 10.
Fig. 13. Endmember pixels extracted by N-FINDR with MNF in two
different runs. (a) VD = 9. (b) SSE = 10.
p = 9. These interesting experiments demonstrated one important observation. That is, a larger p = 10 did not guarantee to
perform better than a smaller p = 9. This is mainly due to two
major reasons. One is that unlike an SQEEA which can take
advantage of endmembers extracted for a smaller value of p as
CHANG et al.: NEW GROWING METHOD FOR SIMPLEX-BASED ENDMEMBER EXTRACTION
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Fig. 15. Results of SGA using MNF. (a) VD = 9. (b) SSE = 10.
Fig. 14. Endmember pixels extracted by VCA in two different runs.
(a) VD = 9. (b) SSE = 10.
part of endmembers generated for a larger value of p, the PPI is
an SMEEA and must regenerate all endmembers for different
values of p. The other reason is that the PPI produced different
PPI counts for endmembers in different runs. In this case, the
panel pixel p311 that appeared in one run may also disappear in
another run due to its low PPI count.
Similarly, N-FINDR also extracted different sets of final
endmember pixels in Fig. 13 for both cases, p = 9 and 10.
For VD = 9, among all extracted pixels are only two were
real endmember pixels p311 and p521 representing two distinct
panel signatures p3 and p5 . However, for SSE = 10, three
panel pixels p311 , p412 , and p521 were extracted in one run, and
two panel pixels p311 and p521 were extracted in another run.
Fig. 14 also showed that the VCA suffered from the same
dilemma encountered in the PPI and N-FINDR where only two
panel pixels which might be different were extracted by all the
cases. However, it is also interesting to note that VCA extracted
different sets of distinct panel signatures in two different runs,
{p3 , p5 } and {p3 , p4 } for VD = 9 and SSE = 10, respectively.
Finally, Fig. 15 shows the result produced by our proposed
SGA using the MNF as DR for VD = 9 and SSE = 10 where its
performance was at least comparably to the other three EEAs.
Since SGA is an SQEEA, the set of endmember pixels extracted
by a smaller value of p is always a subset of endmember
pixels extracted by a larger value of p. As a result, endmembers
extracted in Fig. 15(a) for VD = 9 were also endmember pixels
in Fig. 15(b) for SSE = 10. In this particular case, the tenth
extracted endmember pixel occurred to be the panel pixel p412 ,
which represented an additional third panel signature p3 .
Comparing Fig. 15 to Figs. 12–14, the advantages of using
SGA are very obvious. First of all, the final set of endmember
Fig. 16. Endmember pixels extracted by PPI using three different DR transforms and 1000 skewers. (a) VD = 9. (b) SSE = 10.
pixels extracted by the SGA is always identical regardless
of how many runs the SGA is conducted. Second of all, the
endmember pixels extracted for a smaller value of p are always
a subset of endmember pixels extracted for a larger value of p.
Finally, the impact of DR on the performance of EEAs was
also investigated. Figs. 16–18 show that results of the PPI,
N-FINDR, and SGA using three DR transforms for VD = 9 and
SSE = 10. Unlike the synthetic image experiments in Section V
where there was no difference among the three DR transforms
to be used for preprocessing, the results in Figs. 16–18 provided
evidence that it was important to choose an appropriate DR
transform prior to endmember extraction. The experiments
showed that using the ICA as a DR transform yielded the best
performance where all the five distinct panel signatures were
extracted as endmembers. Most importantly, the results also
showed that VD = 9 was a better estimate than SSE = 10. As a
matter of fact, according to Fig. 18, all the five panel signatures
were extracted by the SGA using ICA before the eighth pixel
was produced.
Like Table IV, Table VII also tabulates the computing time
for PPI with 1000 skewers, N-FINDR, VCA, and SGA with
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Fig. 19. Spatial positions of five pure pixels corresponding to minerals:
alunite (A), buddingtonite (B), calcite (C), kaolinite (K), and muscovite (M).
Fig. 17. Endmember pixels extracted by N-FINDR using three different DR
transforms. (a) VD = 9. (b) SSE = 10.
TABLE VIII
VD ESTIMATES FOR THE CUPRITE SCENE IN FIG. 19 WITH
VARIOUS FALSE ALARM PROBABILITIES
B. AVIRIS Cuprite Data
Fig. 18. Endmember pixels extracted by SGA using three different DR
transforms. (a) VD = 9. (b) SSE = 10.
TABLE VII
COMPUTING TIME OF N-FINDR, VCA, AND SGA
VD = 9 for the same computer environment in Table V where
once again the PPI and VCA ran about five times faster than
did the SGA and the SGA was five times faster than was
N-FINDR. As indicated previously, this is because SGA calculated simplex volumes and required much more computing
time than VCA did, which only calculated projections.
Another real image to be used for experiments is a wellknown AVIRIS image scene, Cuprite, NV, shown in Fig. 19,
which has been widely used to study endmember extraction
extensively. It is available online [19] and was collected by
224 spectral bands with 10-nm spectral resolution over the
Cuprite mining site, in 1997. The image has size of 350 ×
350 pixels and is well understood mineralogically where bands
1–3, 105–115, and 150–170 have been removed prior to the
analysis due to water absorption and low SNR in those bands.
As a result, a total of 189 bands were used for experiments.
The ground truth also provides the spatial locations of the five
minerals, Alunite (A), Buddingtonite (B), Calcite (C), Kaolinite
(K), and Muscovite (M) circled and labeled by A, B, C, K,
and M, respectively, which can be used to verify endmembers
extracted by an EEA.
The VD estimated for this image scene was tabulated in
Table VIII with various false alarm probabilities.
For our experiments, VD was chosen to be 22. The selection
of p = 22 is empirical based on the false alarm fixed at probabilities PF = 10−4 . Also, the value of p estimated by the SSE
was 28. The experiments were conducted based on these two
values, i.e., p = 22 and 28 for comparative analysis.
Experiments similar to those conducted for HYDICE image
scene in Section VI-A were also performed for the Cuprite
image scene in Fig. 19.
First of all, Figs. 20–22 demonstrate a dilemma encountered
in the PPI, N-FINDR, and VCA, which is that final results
produced by these three algorithms in two different runs were
inconsistent for both VD = 22 and SSE = 28. In these figures,
the pixels marked by open circles were extracted by algorithms,
CHANG et al.: NEW GROWING METHOD FOR SIMPLEX-BASED ENDMEMBER EXTRACTION
Fig. 20. Endmember pixels extracted by PPI in two different runs.
(a) VD = 22. (b) SSE = 28.
Fig. 21. Endmember pixels extracted by N-FINDR in two different runs.
(a) VD = 22. (b) SSE = 28.
and the pixels marked by the lower cases of “a, b, c, k, m” with
triangles were the desired endmember pixels corresponding to
the five ground truth mineral endmembers provided by [19] and
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Fig. 22. Endmember pixels extracted by VCA in two different runs.
(a) VD = 22. (b) SSE = 28.
marked by the upper cases of “A, B, C, K, M” with yellow
crosses “x” in the sense of spectral similarity measured by the
Spectral Angle Mapper (SAM) [1], [10]. Since similar tables
using the SAM for signature matching can be obtained in the
exact same manner that was performed in [17], the results are
not included here to avoid replication. Therefore, in all the
following experiments, only spatial locations of the extracted
endmember pixels were identified and shown in the figures. Additionally, the numerals in open parentheses underneath these
figures indicate the numbers of extracted endmember pixels
that were identified in correspondence to ground truth mineral
pixels by the SAM. Since the VCA is an SQEEA, the pixels in
Fig. 22 labeled by numbers indicate the order that these pixels
were extracted in sequence by VCA. It should be also noted that
the PPI implemented here used 500 skewers for computational
convenience.
As shown in Figs. 20 and 21, the PPI and the N-FINDR
extracted different numbers of ground truth corresponding
endmember pixels, 5 and 4 in two different runs for both
cases, VD = 22 and SSE = 28. It was also true for the
VCA with SSE = 28. Moreover, these extracted ground truthcorresponding endmember pixels were generally not the same.
This is mainly due to the fact that there are other pixels whose
spectral signatures are also similar and very close to ground
truth mineral signatures. As a result, a different run is very
much likely to extract different pixels corresponding to ground
truth pixels. This evidence was demonstrated in Fig. 22(a)
where the two sets of three endmember pixels corresponding
to Alunite (A), Buddingtonite (B), Calcite (C) produced by
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Fig. 23. Endmember pixels extracted by PPI using three DR transforms using 500 skewers. (a) VD = 22. (b) SSE = 28.
Fig. 24. Endmember pixels extracted by N-FINDR using three DR transforms. (a) VD = 22. (b) SSE = 28.
CHANG et al.: NEW GROWING METHOD FOR SIMPLEX-BASED ENDMEMBER EXTRACTION
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Fig. 25. Endmember pixels extracted by SGA using three DR transforms. (a) VD = 22. (b) SSE = 28.
Fig. 26. Endmember pixels extracted by VCA. (a) VD = 22. (b) SSE = 28.
the VCA in Fig. 22(a) for VD = 22 in two separate runs
were different.
So far, we have demonstrated a major drawback of the use of
randomly generated initial endmembers. Next, we conducted
a comparative analysis and performance evaluation among
the four EEAs, two SMEEAs (PPI and N-FINDR) and two
SQEEAs (VCA and SGA). Since all the four studied EEAs
required DR as a preprocessing, three DR transforms used in
previous experiments were also used for the following experiments. Figs. 23–25 show the results of endmembers extracted
by PPI with 500 skewers, N-FINDR, SGA using three DR
transforms, PCA, MNF, and ICA for VD = 22 and SSE = 28,
respectively, where notations used in these figures are the same
as those used in Figs. 20–22.
As noted earlier, since the VCA has its built-in DR transform,
no experiments on these three DR transforms conducted for
VCA. Fig. 26 also showed the results obtained by the VCA for
VD = 22 and SSE = 28.
By examining results in Figs. 23–26, the PPI, N-FINDR,
SGA, and VCA could extract all desired five pixels that corresponded to ground truth pixels in terms of spectral similarity
measured by the SAM for both VD = 22 and SSE = 28. On
some occasions, there were four endmembers extracted such as
Fig. 24(a) with the PCA and Fig. 25(a) and (b) with the MNF.
However, these could be corrected by ICA, which consistently
yielded the best results in all cases. Interestingly, Fig. 25(a)
and (b) also showed that when the SGA was implemented
with the MNF, only four ground truth corresponding pixels
were extracted even SSE used a higher value of p = 28 than
22 estimated by the VD. This implied that according to our
experiments, the SSE seemed to overestimate the number of
endmembers, 28 which was higher than 22 estimated by VD.
Most importantly, all experiments conducted above demonstrated that our proposed SGA performed at least comparably
with other EEAs with one major advantage, which other EEAs
do not have, consistent final set of selected endmembers.
Finally, Table IX also tabulates the computing time for PPI
with 500 skewers, N-FINDR, VCA, and SGA with VD = 22
for the same computer environment in Table V. As we can
see, the projection-based algorithms PPI and VCA showed
significant savings in computing time compared to simplex
volume-based algorithms SGA and N-FINDR. Nevertheless,
the SGA was five times faster than the N-FINDR.
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TABLE IX
COMPUTING TIME OF N-FINDR, VCA, AND SGA
VII. C ONCLUSION
This paper introduces a new simplex growing method, an
SGA that resolves three major issues encountered in the
N-FINDR: 1) determination of the number of endmembers;
2) inconsistent final selection set of endmembers; and 3) computational complexity. Coincidentally, the idea of growing simplexes to find endmembers used in the SGA is similar to that
used in a recently developed VCA, but their approaches are
different. Most distinctly, the VCA still cannot address the
second issue, which is inconsistent results in finding a desired
set of endmembers due to its use of random projections for
simplexes generated by zero-mean Gaussian distributions as a
simplex grows.
As a final concluding remark, there is a commercial version
of the N-FINDR that is available [20] which also resolves the
second and the third issues. Unfortunately, the details of its
implementation are not made available to users. In particular,
it does not allow users to select their own initial endmembers
for comparison. Therefore, it was not studied in this paper.
Nevertheless, computational complexity given in Table I remains the same. Additionally, since it is believed to be written
in C not MATLAB, the computation is expected to be much
faster than the one using MATLAB in this paper. It is our belief
that if our proposed SGA is also written in C, its computation
can be significantly improved and comparable to the commercial version of the N-FINDR.
ACKNOWLEDGMENT
The authors would like to thank Dr. Nascimento who provided his developed VCA to be implemented in this paper. The
authors would also like to thank the anonymous reviewers who
suggested the use of a different DR transform for comparative
analysis and to provide the reference [14].
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[20] A. J. Plaza, P. Martinez, A. Plaza and R. M. Perez “Quantifying the
impact of spatial resolution on endmember extraction from hyperspectral
imagery,” in Proc. 3rd EARSeL Workshop Imag. Spectrosc., Germany,
2003, pp. 117–125.
Chein-I Chang (S’81–M’82–SM’92) received the
B.S. degree in mathematics from Soochow University, Taipei, Taiwan, R.O.C., in 1973, the M.S. degree
in mathematics from the Institute of Mathematics,
National Tsing Hua University, Hsinchu, Taiwan,
R.O.C., in 1975, the M.A. degree in mathematics
from the State University of New York, StonyBrook,
in 1977, the M. S. and M.S.E.E. degrees from the
University of Illinois, Urbana-Champaign, in 1982,
and the Ph.D. degree in electrical engineering from
the University of Maryland, College Park, in 1987.
Since 1987, he has been with UMBC as a Visiting Assistant Professor from
January 1987 to August 1987, Assistant Professor from 1987 to 1993, Associate
Professor from 1993 to 2001, and Professor in the Department of Computer
Science and Electrical Engineering since 2001. From 1994 to 1995, he was
a Visiting Research Specialist in the Institute of Information Engineering at
the National Cheng Kung University, Tainan, Taiwan. He was a Distinguished
Lecturer at the National Chung Hsing University, Taichung, Taiwan, sponsored
by the Ministry of Education, from 2005 to 2006. He was on the Editorial Board
of the Journal of High-Speed Networks and was the Guest Editor of a special
issue of the same journal on telemedicine and applications. He is the author of
Hyperspectral Imaging: Techniques for Spectral Detection and Classification
(New York: Kluwer, 2003). He has three patents and several pending patents.
His research interests include multispectral/hyperspectral image processing,
automatic target recognition, medical imaging, information theory and coding,
signal detection and estimation, and neural networks.
Dr. Chang received a National Research Council Senior Research Associateship Award from 2002 to 2003, sponsored by the U.S. Army Soldier and
Biological Chemical Command, Edgewood Chemical and Biological Center.
He is an Associate Editor in the area of hyperspectral signal processing for
the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. He is a
Fellow of the International Society for Optical Engineering and a member of
Phi Kappa Phi and Eta Kappa Nu.
CHANG et al.: NEW GROWING METHOD FOR SIMPLEX-BASED ENDMEMBER EXTRACTION
Chao-Cheng Wu (S’04) received the B.S. degree
in electrical engineering from Tamkang University,
Taipei, Taiwan, R.O.C., in 2002 and the M.S. degree in electrical engineering from the University
of Maryland, Baltimore County (UMBC), Baltimore,
in 2006.
He is currently a Research Assistant in the Remote
Sensing, Signal, and Image Processing Laboratory
at UMBC. His research interests include signal and
hyperspectral image processing.
Wei-min Liu (S’03) received the B.S. degree in
applied mathematics from the National Chiao Tung
University, Taiwan, R.O.C., in 2001, and the M.S.
degree in electrical engineering from the University
of Maryland, Baltimore County (UMBC), Baltimore,
in 2005. He is currently working toward the Ph.D.
degree at UMBC.
He is currently a Research Assistant in the
Remote Sensing, Signal, and Image Processing
Laboratory, UMBC. His research interests include
multi/hyperspectral and medical image processing,
pattern recognition, and adaptive signal processing.
2819
Yen-Chieh Ouyang (S’86–M’92) received the
B.S.E.E. degree from Feng Chia University,
Taichung, Taiwan, in 1981, and the M.S. and
Ph.D. degrees, both from University of Memphis,
Memphis, Tennessee, in 1987, and 1992, respectively, all in electrical engineering.
He joined the faculty of the Department of
Electrical Engineering at National Chung Hsing
University (NCHU), Taichung, Taiwan, R.O.C., in
August 1992. He is currently an Associate Professor
and the Director of the Multimedia Communication Laboratory, NCHU. His research interests include hyperspectral image
processing, communication networks, providing QoS over wireless and network security in mobile networks, multimedia system design, and performance
evaluation.